4.1 Q-8

Question Statement

The points $A(-5, -2)$ and $B(5, -4)$ are the ends of a diameter of a circle. Find the center and radius of the circle.

Background and Explanation

In a circle, the center lies at the midpoint of the diameter. The radius is half the length of the diameter. We can use the distance formula to calculate the length of the diameter between the two points, then divide by 2 to find the radius. The midpoint formula helps find the center of the circle by determining the average of the x and y coordinates of the two points.

Midpoint Formula: The midpoint of two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Solution

Step 1: Find the Center of the Circle

The center of the circle is the midpoint of the diameter, which is the midpoint of points $A(-5, -2)$ and $B(5, -4)$ .

Using the midpoint formula:

\text{Midpoint} = \left( \frac{-5 + 5}{2}, \frac{-2 + (-4)}{2} \right)

Simplifying:

\text{Midpoint} = \left( \frac{0}{2}, \frac{-6}{2} \right) = (0, -3)

Thus, the center of the circle is at $(0, -3)$ .

Step 2: Find the Radius of the Circle

The radius is half the length of the diameter. We can first find the length of the diameter using the distance formula.

The coordinates of the points $A(-5, -2)$ and $B(5, -4)$ are given. To calculate the distance (diameter), apply the distance formula:

AB = \sqrt{(5 - (-5))^2 + (-4 - (-2))^2}

Simplifying:

AB = \sqrt{(5 + 5)^2 + (-4 + 2)^2}

AB = \sqrt{(10)^2 + (-2)^2} = \sqrt{100 + 4} = \sqrt{104}

Now, calculate the radius:

\text{Radius} = \frac{1}{2} \times AB = \frac{1}{2} \times \sqrt{104}

Simplifying further:

\text{Radius} = \frac{1}{2} \times \sqrt{4 \times 26} = \frac{1}{2} \times 2 \sqrt{26} = \sqrt{26}

Thus, the radius of the circle is $\sqrt{26}$ .

Key Formulas or Methods Used

Midpoint Formula: The midpoint of two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Summary of Steps

Find the center: Use the midpoint formula to calculate the midpoint of points $A(-5, -2)$ and $B(5, -4)$ , giving the center as $(0, -3)$ .
Find the diameter: Use the distance formula to find the length of the diameter between points $A$ and $B$ .
Find the radius: The radius is half the length of the diameter, so divide the diameter by 2. The radius is $\sqrt{26}$ .